Problem 50 - College Algebra

Question

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\left\\{\frac{1}{3},-\frac{4}{3},-3, \ldots\right\\} $$

Step-by-Step Solution

Answer

The explicit formula is \( a_n = 2 - \frac{5}{3}n \).

Step 1: Identify Common Difference

To find the explicit formula for an arithmetic sequence, we start by identifying the common difference between consecutive terms. In the given sequence \( \left\{ \frac{1}{3}, -\frac{4}{3}, -3, \ldots \right\} \), we subtract the first term from the second: \( -\frac{4}{3} - \frac{1}{3} = -\frac{5}{3} \). We check by subtracting the second from the third: \( -3 - (-\frac{4}{3}) = -\frac{5}{3} \). Thus, the common difference \( d \) is \(-\frac{5}{3}\).

Step 2: Identify First Term

The first term of the sequence is given as \( a_1 = \frac{1}{3} \). This is crucial for writing the explicit formula.

Step 3: Construct Explicit Formula

The explicit formula for an arithmetic sequence is given by \( a_n = a_1 + (n-1) \cdot d \). Substituting the values we found, \( a_1 = \frac{1}{3} \) and \( d = -\frac{5}{3} \), the formula becomes \( a_n = \frac{1}{3} + (n-1) \cdot \left(-\frac{5}{3}\right) \).

Step 4: Simplify the Formula

Simplifying the formula \( a_n = \frac{1}{3} + (n-1)\left(-\frac{5}{3}\right) \), we expand it to get \( a_n = \frac{1}{3} - \frac{5}{3}(n-1) \). Further simplifying gives us \( a_n = \frac{1}{3} - \frac{5}{3}n + \frac{5}{3} \). Combining like terms results in \( a_n = \frac{6}{3} - \frac{5}{3}n \) or \( a_n = 2 - \frac{5}{3}n \).

Key Concepts

Common Difference in Sequences

When dealing with arithmetic sequences, one key aspect is understanding the common difference. This is the amount you add or subtract to go from one term in the sequence to the next. For our given sequence, \( \left\{ \frac{1}{3}, -\frac{4}{3}, -3, \ldots \right\} \), finding the common difference involves subtracting consecutive terms.

For example:

• Subtract the first term from the second: \( -\frac{4}{3} - \frac{1}{3} = -\frac{5}{3} \).
• Confirm by checking the next pairs: \( -3 - (-\frac{4}{3}) = -\frac{5}{3} \).

This consistent result shows the common difference, \( d \), is \(-\frac{5}{3}\). Recognizing this pattern is crucial, as it ensures you can construct formulas and predict future terms.

Explicit Formula for Sequences

The explicit formula of an arithmetic sequence provides a way to find any term in the sequence without listing all the previous ones. This formula is especially useful for quickly finding terms far down the line. Generally, the formula is represented as:

\[ a_n = a_1 + (n-1) \cdot d \]Here, \( a_n \) denotes the \( n\)th term, \( a_1 \) is the first term, \( n \) is the term’s position in the sequence, and \( d \) is the common difference.

For our sequence:

• First term, \( a_1 = \frac{1}{3} \)
• Common difference, \( d = -\frac{5}{3} \)

Inserting these into the formula, we have:
\[ a_n = \frac{1}{3} + (n-1) \cdot (-\frac{5}{3}) \]This equation allows you to compute any term of the sequence efficiently.

Sequence Simplification

After constructing the explicit formula, it often needs simplification to make calculations easier or to identify patterns. Start by expanding the terms and then combining like ones.

For example, our formula is initially:
\[ a_n = \frac{1}{3} + (n-1) \cdot \left(-\frac{5}{3}\right) \]After expanding this, you get:
\[ a_n = \frac{1}{3} - \frac{5}{3}n + \frac{5}{3} \]Combine like terms, specifically the constants:
\[ a_n = \frac{6}{3} - \frac{5}{3}n = 2 - \frac{5}{3}n \]This clean expression makes it straightforward to find specific terms and recognize trends or behaviors in the sequence. Simplification is not just about tidying up the formula; it can expose insights and make computations faster.